f(t) dt, x > 0, is the best value and it is the norm of the

Transcription

1 MATEMATIQKI VESNIK 66, 1 (214), March 214 originalni nauqni rad research aer GENERALIZED HAUSDORFF OPERATORS ON WEIGHTED HERZ SPACES Kuang Jichang Absrac. In his aer, we inroduce new generalized Hausdorff oeraors. They include many famous oeraors as secial cases. We obain necessary and sufficien condiions for hese oeraors o be bounded on he weighed Herz saces. The corresonding new oeraor norm inequaliies are obained. They are significan imrovemens and generalizaions of many known resuls. Several oen roblems are formulaed. 1. Inroducion The classical Hardy oeraor T is defined by T (f, x) = 1 x x f() d, x >, and he classical Hardy inequaliy saed in Hardy e al. [4] is T f where he bes consan facor oeraor T, ha is, 1 f, 1 < <, (1.1) 1 T = is he bes value and i is he norm of he ( 1). As oined ou by Kufner e al. [6], he Hardy inequaliy has a fascinaing as and will have (hoefully) also a fascinaing fuure. These auhors of [6] resen some imoran ses of he develomen of (1.1), of is early weighed generalizaions and of is various modificaions and exensions. Anoher classical oeraor is he Hausdorff oeraor ψ() ( x ) T 1 (f, x) = f d, x >, (1.2) 21 AMS Subjec Classificaion: 26D1, 47A3 Keywords and hrases: Hausdorff oeraor; weighed Herz sace; norm inequaliy. 19

2 2 K. Jichang where ψ is a local inegrable funcion in (, ). This oeraor and is varieies have araced many auhors, for examle, see [7 1]. Wih u = x, (1.2) yields ψ( x u T 1 (f, x) = ) f(u) du, x >. u Choosing ψ(u) = u 1 ϕ E1 (u) and ψ(u) = ϕ E2 (u), where E 1 = (1, ), E 2 = (, 1], and ϕ E denoes he characerisic funcion of he se E, we obain he Hardy oeraor T and he dual Hardy oeraor (or Cesáro oeraor) T defined by T f(u) (f, x) = du, x >, x u resecively. In addiion, T 2 = T +T becomes he Calderòn maximal oeraor [1]: T 2 (f, x) = 1 x f() f() d + d, x >. x x The aim of his aer is o inroduce he following new generalized Hausdorff oeraor ψ() T (f, x) = f(g()x) d, x = (x 1, x 2,..., x n ) R n, (1.3) where g()x = (g()x 1, g()x 2,..., g()x n ), ψ : (, ) (, ) is a locally inegrable funcion, g : (, ) (, ) is a monoonic funcion (increasing or decreasing), and f is a measurable comlex valued funcion on R n. If g() = 1 and n = 1, hen T reduces o he Hausdorff oeraor T 1. If we inroduce oher forms of g and ψ, i is ossible o obain oher oeraors of ineres. For examle, if g() =, ψ() = ω()ϕ E (), where E = (, 1], ω is a non-negaive weigh funcion, hen T reduces o he weighed Hardy-Lilewood mean oeraor defined in [12]: T 3 (f, x) = 1 f(x)ω() d, x = (x 1, x 2,..., x n ) R n. (1.4) If g() = 1, ψ() = (n 1) ω()ϕ E (), ω, E are as in (1.4), hen T reduces o he weighed Cesàro mean oeraor defined in [12]: 1 ( x ) T 4 (f, x) = f n ω() d, x = (x 1, x 2,..., x n ) R n. If g() =, ψ() = e λ, λ >, n = 1, x >, hen T reduces o T 5 (f, x) = f(x)e λ d = 1 f(u)e αu du = 1 L(f, α), x x where L(f, α) = f(u)e αu du is he Lalace ransform of f, u = x, α = λ x >. Hence, (1.3) is a significan generalizaion of many famous oeraors. I is well-known ha he Herz saces lay an imoran role in characerizing he roeries of funcions and muliliers on he classical Hardy saces. In his aer, we obain necessary and sufficien condiions for he generalized Hausdorff oeraor T defined by (1.3) o be bounded on he weighed Herz saces. The corresonding new oeraor norm inequaliies are obained. Several oen roblems are formulaed.

3 Generalized Hausdorff oeraors Definiions and saemen of he main resuls Le k Z, B k = {x R n : x 2 k }, D k = B k B k 1 and le ϕ k = ϕ Dk denoe he characerisic funcion of he se D k. Moreover, for a measurable funcion f on R n and a non-negaive weigh funcion ω(x), we wrie ( ) 1/ f,ω = f(x) ω(x) dx. R n In wha follows, if ω 1, hen we will denoe L (R n, ω) (in brief L (ω)) by L (R n ). Definiion 2.1. (see [11]) Le α R 1, <, q < and ω 1 and ω 2 be nonnegaive weigh funcions. The homogeneous weighed Herz sace K q (ω 1, ω 2 ) is α, defined by where K α, q (ω 1, ω 2 ) = {f L q loc (Rn {}) : f K q α, (ω 1,ω 2 ) < }, { } 1/. f K q α, (ω 1,ω 2 ) = [ω 1 (B k )] α n fϕ k q,ω 2 We can similarly define he non-homogeneous weighed Herz saces K α, q (ω 1, ω 2 ). I is easy o see ha when ω 1 = ω 2 = 1, we have K q α, (1, 1) = K α, q (R n ), K (α/), (R n ) = L ( x α dx), K, (R n ) = L (R n ). Definiion 2.2. (see [2]) A non-negaive weigh funcion ω saisfies Muckenhou s A condiion or ω A, if here is a consan C indeenden of he cube Q in R n, such ha ( 1 Q Q ) ω(x) dx where Q is he Lebesgue measure of Q. { ( ) } 1 1 ex log dx C, Q R n, Q Q ω(x) Our main resuls are he following hree heorems: Theorem 2.1. Le α R 1, < <, 1 q <, ω 1 A, and a non-negaive weigh funcion ω 2 saisfy ω 2 (x) = β ω 2 (x), >, β R 1, x R n (2.1) Le ψ : (, ) (, ) be a locally inegrable funcion having he comac suor on (, ); le g : (, ) (, ) be an increasing funcion saisfying he submulilicaive condiion g(uv) g(u)g(v), u, v >. Le T be he norm of he oeraor T defined by (1.3) and maing K q α, α, (ω 1, ω 2 ) K q (ω 1, ω 2 ).

4 22 K. Jichang where (1) If g() (β+n)/q ψ() g() αδ (β+n)/q ψ() C(, α) = is a concave funcion on (, ) and d <, hen T C(, α) αδ (β+n)/q ψ() g() d, (2.2) { C α n 2(1/) 2 (1 + ) 1/ (1 + g(2) α δ ), < < 1, C α n 2 1 (2/) (1 + (1/))(1 + g(2) α δ ), 1 <. (2.3) (2) If T <, and g is a sricly increasing funcion on (, ) and he inverse g 1 of g saisfies g 1 () ( + ), hen αδ (β+n)/q ψ() g() d T. (C and δ are consans given in (3.4), see Secion 3 below.) Remark 1. ω 2 is an exension of he ower weigh ω 2 (x) = x β, (x R n ). We use he following noaion: { ( α, ψ() ) KF = f : f K q (ω 1, ω 2 ), F () = su f(g()x) x R n is a concave funcion on (, ) }. Then KF is a subsace of he sace K α, q (ω 1, ω 2 ). Theorem 2.2 Le α R 1, < <, < q < 1, g, ψ, ω 1, ω 2 be as in Theorem 2.1, and T be he norm of he oeraor T defined by (1.3), maing α, KF K q (ω 1, ω 2 ). (1) If g() (β+n)/q ψ() g() αδ (β+n)/q ψ() where C(, q, α) = is a concave funcion on (, ) and d <, hen T C(, q, α, ) αδ (β+n)/q ψ() g() d, (2.4) C α/n 2 (1/) (1/q) 2 q 1/ (1+q) 1/q (+q) 1/ (1+g(2) α δ ), < q < 1, C α/n 2 (1/q) 2 (1 + q) 1/q (1 + g(2) α δ ), < q < < 1, C α/n 2 (1/q) (2/) 1 (1+q) 1/q (1+(1/))(1+g(2) α δ ), <q<1 <. (2.5) (2) If T <, and g is a sricly increasing funcion on (, ) and he inverse g 1 of g saisfies g 1 () ( + ), hen αδ (β+n)/q ψ() g() d T. (C and δ are consans given in (3.4), see Secion 3 below.)

5 Generalized Hausdorff oeraors 23 Remark 2. When g is a decreasing funcion, similar resuls wih g(2 1 ) and g 1 () ( ) insead of g(2) and g 1 () ( ) can be obained from he revious wo heorems. Theorem 2.3. Le β R 1, 1 <, and g, ψ be wo osiive measurable funcions defined on (, ), and g be a sricly monoonic funcion on (, ) and he inverse g 1 of g saisfies: (1) when g is increasing, g 1 () ( + ); (2) when g is decreasing, g 1 () ( ), and a nonnegaive weigh funcion ω saisfies ω(x) = β ω(x), >, β R 1, x R n. Then he oeraor T defined by (1.3), maing L (ω) L (ω), exiss as a bounded oeraor if and only if (β+n)/ ψ() g() d <. (2.6) Moreover, when (2.6) holds, he oeraor norm T of T on L (ω) saisfies T = g() (β+n)/ ψ(). (2.7) Remark 3. I follows from (2.7) and (1.3) ha } ψ() 1/ f(g()x) d R n ω(x) dx ( ) } 1/ (β+n)/ ψ() g() d f(x) ω(x) dx, (2.8) R n where T = g() (β+n)/ ψ() d is he bes ossible consan. In aricular, when ψ() = λ ϕ E (), E = (, 1], g() =, n = 1, and (, ) is subsiued by (, ), and λ >, β < λ 1, hen by (2.7), we ge T = λ (β + 1). I follows from (2.8) ha { 1 x 1/ x λ f() λ 1 d 1/ ω(x) dx} f(x) ω(x) dx}, λ (β + 1) (2.9) where T = λ (β+1) is he bes ossible consan. If λ = 1, ω(x) = 1, ha is, β =, hen (2.9) reduces o he Hardy inequaliy (1.1). If λ = 1, ω(x) = x β, hen (2.9) reduces o he resul of [6,. 23]. When ψ() = λ ϕ E (), E = (, 1], λ, g() = 1, n = 1, and (, ) is subsiued by (, ), and β > λ 1, hen by (2.7), we ge T = ( 1 x λ x (β+1) λ f() λ 1 d ω(x) dx) 1/. I follows from (2.8) ha ( 1/ f(x) ω(x) dx), (β + 1) λ

6 24 K. Jichang where T = (β+1) λ is he bes ossible consan. This is he weighed exension of he dual Hardy inequaliy in [5,6]. When g() =, ψ() = e λ, λ >, n = 1, and (, ) is subsiued by (, ), and β < 1, x >, hen by (2.7), we ge T = (β+1)/ e λ d = Γ(1 (β + 1)/) λ 1 (β+1)/. I follows from (2.8) ha { 1 } 1/ f()e (λ/x) d ω(x) dx x Γ(1 (β + 1)/) 1/ f(x) ω(x) dx}, (2.1) λ 1 (β+1)/ where T = Γ(1 (β+1) ) (β+1) 1 λ is he bes ossible consan. In aricular, when f(x), ω(x) = 1, ha is, β =, hen (2.1) reduces o he following Lalace ransform inequaliy: { ( } 1/ f()e d) (λ/x) x dx ( Γ(1 (1/)) 1/ f (x) dx). (2.11) λ 1 (1/) Remark 4. Hardy [4, Theorems 35, 352] roved he following hree inequaliies: Le 1 < <, q = 1, f(x), ω(x) = x 2. Then L(f) Γ(1/) f,ω, (2.12) L(f),ω Γ(1 1 ) f. (2.13) For 1 < 2, L(f) q ( 2π q )1/q f. (2.14) These inequaliies are he Lalace ransforms analogues of inequaliies in he heory of Fourier series. As oined ou by Hardy [3,4], i is no assered ha he consan in (2.14) is he bes ossible and i may be difficul o find he bes ossible value. Here we rove ha he consans in (2.8) (2.11) are he bes ossible. Hence, our main resuls are significan generalizaion of many known resuls. Remark 5. There are some similar resuls for he non-homogeneous weighed Herz saces Kq α, (ω 1, ω 2 ). We omi he deails here. Several oen roblems. In Theorem 2.3, we solve he bes value for C(, β) in he following inequaliy T f,ω C(, β) f,ω,

7 Generalized Hausdorff oeraors 25 ha is, C(, β) = T = g() (β+n)/ ψ() d is he bes ossible consan, bu ye in Theorems 2.1 and 2.2, he bes value for C(, α) and C(, q, α) in he following inequaliies T f K C(, α) f K and T f K C(, q, α) f KF are no solved. I is no assered ha he consans C(, α) and C(, q, α) in Theorems 2.1 and 2.2 are he bes ossible. 3. Proofs of he heorems We require he following lemmas o rove our resuls. Lemma 3.1. Le f be a nonnegaive measurable funcion on [, b], < b <. If 1 <, hen ( ) f(x) dx b ( 1) f (x) dx. (3.1) Lemma 3.1 is an immediae consequences of Hölder inequaliy. Lemma 3.2. (see [5]) Le f be a nonnegaive measurable and concave funcion on [a, b], < α β. Then { β + 1 b 1/β { α + 1 b 1/α [f(x)] dx} β [f(x)] dx} α. (3.2) b a b a a Seing a =, for α =, β = 1, ha is, < 1, we obain from (3.2) ( ) f(x) dx b 1 f (x) dx (3.3). By he roeries of A weighs, we have Lemma 3.3 (see [2]) If ω A, hen here exis δ >, C >, such ha for each ball B in R n and measurable subse E of B, ( ) δ ω(e) E ω(b) C, (3.4) B where E is he Lebesgue measure of E and ω(e) = ω(x) dx. E Lemma 3.4 (see [5]) (C inequaliy) Le a 1, a 2,..., a n be arbirary real (or comlex) numbers. Then ( n a k k=1 ) C ( n a a k ), < <, (3.5) k=1

8 26 K. Jichang where { 1, < < 1, C = n 1, 1 <. In wha follows, we shall wrie simly denoe B k = {x R n : x 2 k }. α, K q (ω 1, ω 2 ) o denoe K and B(2 k ) o Proof of Theorem 2.1. Since ψ has a comac suor on (, ), here exiss b >, such ha su ψ() (, b]. Firs, we rove (2.2). Using Minkowski s inequaliy for inegrals and (2.1), and seing u = g()x, we ge } 1/q (T f)ϕ k q,ω2 f(g()x) q ψ() ω 2 (x) dx d D k = f(u) q ω 2 (u) du 2 k 1 g()< u 2 k g() } 1/q g() (β+n)/q ψ() For each (, b), here exiss an ineger m such ha 2 m 1 < 2 m. Seing we obain A 1 = {u R n : 2 k 1 g(2 m 1 ) < u 2 k g(2 m 1 )}, E 1 = B(2 k g(2 m 1 )), A 2 = {u R n : 2 k g(2 m 1 ) < u 2 k g(2 m )}, E 2 = B(2 k g(2 m )), I follows ha (T f)ϕ k q,ω2 T f K { [ω 1 (B k )] α n [ f(u) q ω 2 (u) du A 1 } 1/q + f(u) q (β+n)/q ψ() ω 2 (u) du g() A 2 Now, we consider wo cases for : (β+n)/q ψ() ( fϕ A1 q,ω2 + fϕ A2 q,ω2 )g() d. } 1/ (β+n)/q ψ() ( fϕ A1 q,ω2 + fϕ A2 q,ω2 )g() d]. (3.6) Case 1. < < 1. In his case, i follows from (3.6), (3.3) and (3.5) ha { (1 + )1/ T f K [ω 2b (1/) 1 1 (B k )] α/n ( ) 1/ ψ() ( fϕ A1 q,ω 2 + fϕ A2 q,ω 2 )g() d} (β+n)/q {[ ( 2 (1/) 2 (1 + ) 1/ b 1 ( 1 ) ω 1 (E 1 ) α/n fϕ A1 ω1 (B k ) q,ω 2 ω 1 (E 1 ) d d. ) α/n

9 Generalized Hausdorff oeraors 27 g() (β+n)/q ( ψ() 1/ [ ) d] + ( ω1 (B k ) ω 1 (E 2 ) k (Z) ω 1 (E 2 ) α/n fϕ A2 q,ω 2 ) α/n g() (β+n)/q ( ψ() ) d] 1/ }. (3.7) By (3.4) and B k = πn/2 Γ( n 2 +1) 2kn, we have ( ) δ ω 1 (B k ) ω 1 (E 1 ) C Bk = C g(2 m 1 ) nδ (3.8) E 1 and ω 1 (B k ) ω 1 (E 2 ) C g(2 m ) nδ. (3.9) I follows from (3.7) (3.9) ha T f K C α n 2 (1/) 2 (1 + ) 1/ f K (g(2 m 1 ) αδ + g(2 m ) αδ (β+n)/q ψ() )g() d. (3.1) By he submulilicaiviy of g, we have g(2 m ) g(2)g(2 m 1 ). If α >, hen [g(2 m 1 )] αδ [g(2 m )] αδ [g(2)] αδ. Since g is a increasing funcion, hus 2 m imlies ha g() g(2 m ), and herefore ha g(2 m ) αδ g() αδ. This imlies g(2 m 1 ) αδ + g(2 m ) αδ g(2 m ) αδ {1 + g(2) αδ } g() αδ {1 + g(2) αδ }. (3.11) Similarly, if α <, hen (3.11) and (3.12) imly ha g(2 m 1 ) αδ + g(2 m ) αδ g() αδ {1 + g(2) αδ }. (3.12) g(2 m 1 ) αδ + g(2 m ) αδ g() αδ {1 + g(2) α δ }. (3.13) Thus, by (3.1) and(3.13), we ge Tf K C α/n 2 (1/) 2 (1+) 1/ (1+g(2) α δ ) f K g() αδ (β+n)/qψ() d. (3.14) Case 2. 1 <. In his case, by (3.6), (3.1), (3.3), (3.8) and (3.9), we similarly obain T f K (2b) 1 (1/) { [ω 1 (B k )] α n ( ) 1/ ψ() ( fϕ A1 q,ω 2 + fϕ A2 q,ω 2 )g() d} ( (β+n)/q) C α n 2 1 (2/) (1 + (1/))(1 + g(2) α δ ) f K αδ (β+n)/q ψ() g() d. (3.15)

10 28 K. Jichang Hence, by (3.14) and (3.15), we ge T C(, α) where C(, α) is defined by (2.3). αδ (β+n)/q ψ() g() d, To rove he oosie inequaliy, uing ε (, 1), we se ω 1 (B k ) = 2 knδ, ω 2 (x) = x β, and {, x 1, f ε (x) = x αδ (β+n)/q ε, x > 1. Then for k =, 1, 2,..., f ε ϕ k q,ω2 = and for k Z +, we have ( ) 1/q f ε ϕ k q,ω2 = x (αδ+(β+n)/q+ε)q ω 2 (x) dx 2 k 1 < x 2 k { 2π n/2 2 k } 1/q = r (αδ+ε)q 1 dr = Cn 1/q 2 k(αδ+ε), Γ(n/2) 2 k 1 where C n = 2πn/2 2 (αδ+ε)q 1 Γ(n/2) (αδ + ε)q. I follows ha { ) /q } 1/ f ε K = ω 1 (B k ) ( 2 α/n x (αδ+(β+n)/q+ε)q ω 2 (x) dx k 1 < x 2 k k=1 = C 1/q n { n=1 2 kε } 1/ = C 1/q n 2 ε Since g() x > 1 imlies ha > g 1 ( 1 x ), we have T (f ε, x) = x αδ (β+n)/q ε. (3.16) (1 2 ε ) 1/ g 1 ( 1 x ) g() αδ (β+n)/q ε ψ() Le ε ( < ε < 1) be given. Since g is a sricly increasing funcion, here exiss m N, such ha 2 m 1 1 ε < 2m and 2 mε 1 (ε + ). Noe ha 2 k 1 < x 2 k, so ha if k m + 1, hen g 1 ( 1 x ) g 1 1 ( ) g 1 ( 1 2 k 1 2 ) g 1 (ε). m Thus, T f ε K = ω 1 (B k ) α/n = x >1 ω 1 (B k ) α/n k=1 ( g() g 1 ( 1 x ) } /q [T (f ε, x)ϕ k (x)] q ω 2 (x) dx 2 k 1 < x 2 k x (αδ+(β+n)/q+ε)q αδ (β+n)/q ε ψ() q } /q d) ω 2 (x) dx d.

11 k=m+1 Generalized Hausdorff oeraors 29 ( g() g 1 (ε) αδ (β+n)/q ε ψ() ) d ω 1 (B k ) α/n 2 k 1 < x 2 k x (αδ+(β+n)/q+ε)q ω 2 (x) dx ( = g() g 1 (ε) αδ (β+n)/q ε ψ() Thus, by (3.16) and (3.17), we ge T T f { ε K 2 mε g() f ε K g 1 (ε) Taking limi as ε in (3.18), we obain } /q ) d Cn /q 2 (m+1)ε (1 2 ε ). (3.17) αδ (β+n)/q ε ψ() αδ (β+n)/q ψ() T g() d. This finishes he roof of Theorem 2.1. } d. (3.18) Proof of Theorem 2.2. Firs, we rove (2.4). Using (3.3) and he noaions in he roof of Theorem 2.1, we obain ( (T f)ϕ k q,ω2 ( su f(g()x) ) ψ() q } 1/q d) ω 2 (x) dx D k x R n ( 2 1 (1 + q) 1/q b 1 (1/q) ( su f(g()x) ) q ( ψ() ) } 1/q ) q d ω 2 (x) dx x R n D k 2 1 (1 + q) 1/q b 1 (1/q) { ( f(u) q ω 2 (u) du 2 k 1 g()< u 2 k g() 2 1 (1 + q) 1/q b 1 (1/q) { I follows ha ) g() (β+n) ( ψ() ) q d } 1/q ( fϕ A1 q q,ω 2 + fϕ A2 q q,ω 2 )g() (β+n) ( ψ() 1/q ) d} q. { T f K 2 1 (1 + q) 1/q b 1 (1/q) ω 1 (B k ) α/n [ ( fϕ A1 q q,ω 2 + fϕ A2 q q,ω 2 )g() (β+n) ( ψ() /q } 1/ ) d] q. (3.19) Now,we consider hree cases: Case 1. < q < 1. In his case, i follows from (3.19), (3.3), (3.5), (3.1), (3.8) and (3.9) ha T f K 2 (1+(1/q)) q (1/) (1 + q) 1/q ( + q) 1/ b 1 (1/) { ω 1 (B k ) α/n ( fϕ A1 q,ω 2 + fϕ A2 q,ω 2 )g() (β+n)/q ( ψ() ) d } 1/

12 3 K. Jichang 2 (1/) (1/q) 2 q (1/) (1 + q) 1/q ( + q) 1/ f K { ( ω 1(B k ) ω 1 (E 1 ) )α/n +( ω } 1(B k ) ω 1 (E 2 ) )α/n (β+n)/q ψ() g() d C α/n 2 (1/) (1/q) 2 q (1/) (1 + q) 1/q ( + q) 1/ f K (g(2 m 1 ) αδ +g(2 m ) αδ (β+n)/q ψ() )g() d C α/n 2 (1/) (1/q) 2 q (1/) (1 + q) 1/q ( + q) 1/ (1 + g(2) α δ ) f K αδ (β+n)/q ψ() g() d. (3.2) Case 2. < q < 1. In his case, by (3.19), (3.5), (3.1), (3.8) and (3.9), we similarly obain T f K { (1 + q)1/q b 1 (1/) ω 1 (B k ) α/n 2 ( fϕ A1 q q,ω 2 + fϕ A2 q ) /q g() (β+n)/q ( ψ() { ( ω ( 1(B k ) ω1 (B k ) ω 1 (E 1 ) )α/n + ω 1 (E 2 ) 2 (1/q) 2 (1 + q) 1/q (1 + g(2) α δ ) f K 2 (1/q) 2 (1 + q) 1/q f K C α/n } 1/ ) d ) α/n } (β+n)/q ψ() g() d αδ (β+n)/q ψ() g() d. (3.21) Case 3. < q 1 <. In his case, by (3.19), (3.1), (3.5), (3.3), (3.8) and (3.9), we obain T f K (1 + q)1/q b 1 (1/) { ω 2 (1/)+1 (1/q) 1 (B k ) α/n ( fϕ A1 q,ω 2 + fϕ A2 q,ω 2 )g() (β+n)/q ( ψ() ) d } 1/ C α/n 2 (1/q) (2/) 1 (1 + q) 1/q (1 + (1/))(1 + g(2) α δ ) f K Hence, by (3.2) (3.22), we ge αδ (β+n)/q ψ() g() d. (3.22) αδ (β+n)/q ψ() T C(, q, α) g() d, where C(, q, α) is defined by (2.5). By he same echnique used in Theorem 2.1 one can show ha he oosie inequaliy: αδ (β+n)/q ψ() T g() d. This finishes he roof of Theorem 2.2.

13 Generalized Hausdorff oeraors 31 Proof of Theorem 2.3. By Minkowski inequaliy for inegrals and seing u = g()x, we ge T f,ω ( f(g()x) ψ() } 1/ d) ω(x) dx R n } 1/ f(g()x) ψ() ω(x) dx d R n } 1/ = f(u) (β+n)/ ψ() ω(u) du g() d R n (β+n)/ ψ() = f,ω g() d. I follows ha T f,ω T = su f f,ω (β+n)/ ψ() d. To rove he oosie inequaliy, uing ε (, 1), we se ω(x) = x β and {, x 1, f ε (x) = x (β+n)/ ε, x > 1, hus f ε,ω = x >1 x (β+n+ε) ω(x) dx = 2πn/2 Γ(n/2) 1 r n ε r n 1 dr = 2πn/2 εγ(n/2). If g is sricly increasing, uing x < 1 ε, g() x > 1 imlies ha > g 1 ( 1 x ) > g 1 (ε). I follows ha This imlies T f ε,ω = ( x >1 ( g() g 1 (ε) T T f ε,ω f ε,ω g 1 ( 1 x ) (g() x ) (β+n)/ ε ψ() Taking limis as ε in (3.23), we ge T Then by (3.23) and (3.24), we have T = g 1 (ε) (β+n)/ ε ψ() d ){ 2πn/2 εγ(n/2) }1/. } 1/ d) ω(x) dx (β+n)/ ε ψ() g() d. (3.23) (β+n)/ ψ() g() d. (3.24) (β+n)/ ψ() g() d. The roof for he decreasing case is similar. The heorem is roved.

14 32 K. Jichang Acknowledgemen. The auhor would like o hank anonymous referees on suggesions o imrove his ex. REFERENCES [1] C. Benne, R.A. Devore, R. Sharley, Weak L and BMO, Ann. Mah. 113 (1981), [2] J. García-Cuerva, J.L. Rubio de Francia, Weighed Norm Inequaliies and Relaed Toics, Norh-Holland Publishing, Amserdam, [3] G.H. Hardy, The consans of cerain inequaliies, J. London Mah.Soc. 8 (1933), [4] G.H. Hardy, J.E. Lilewood, G. Polya, Inequaliies (2nd ediion), Cambridge Universiy Press, Cambrige, [5] J.C. Kuang, Alied Inequaliies (4h ediion), Shangdong Science Press, Ji nan, 21 (in Chinese). [6] A. Kufner, L. Maligranda, L.E. Persson, The Hardy inequaliy Abou is hisory and some relaed resuls, Pilsen, 27. [7] A.K. Lerner, E. Liflyand, Mulidimensional Hausdorff oeraors on he real Hardy saces, J. Ausral. Mah. Soc. 83 (27), [8] E. Liflyand, F. Mórecz, The Hausdorff oeraor is bounded on real H 1 sace, Proc. Amer. Mah. Soc. 128 (2), [9] E. Liflyand, Oen roblems on Hausdorff oeraors, in: Comlex Analysis and Poenial Theory (Proceedings of he Conference, Isanbul, Turkey, Se 8 14, 26). [1] E. Liflyand, A. Miyachi, Boundedness of he Hausdorff oeraors in H saces, < < 1, Sudia Mah. 194 (29), [11] S. Lu, D. Yang, The decomosiion of he weighed Herz saces and is alicaions, Sci. in China A 38 (1995), [12] J. Xiao, L and BMO bounds of weighed Hardy Lilewood averages, J. Mah. Anal. Al. 262 (21), (received ; in revised form ; available online ) Dearmen of Mahemaics, Hunan Normal Universiy, Changsha, 4181 P.R.CHINA jc-kuang@homail.com